Wikibooksβ

Problems in Mathematics/To be added

2 ExerciseSuppose f{\displaystyle f} is infinitely differentiable. Suppose, furthermore, that for every x{\displaystyle x}, there is n{\displaystyle n} such that f(n)(x)=0{\displaystyle f^{(n)}(x)=0}. Then f{\displaystyle f} is a polynomial. (Hint: Baire's category theorem.)

Exercisee{\displaystyle e} and π{\displaystyle \pi } are irrational numbers. Moreover, e{\displaystyle e} is neither an algebraic number nor p-adic number, yet ep{\displaystyle e^{p}} is a p-adic number for all p except for 2.

ExerciseThere exists a nonempty perfect subset of R{\displaystyle \mathbf {R} } that contains no rational numbers. (Hint: Use the proof that e is irrational.)

ExerciseProve that a convex function is continuous (Recall that a function f:(a,b)→R{\displaystyle f:(a,b)\rightarrow \mathbb {R} } is a convex function if for all x,y∈(a,b){\displaystyle x,y\in (a,b)} and all s,t∈[0,1]{\displaystyle s,t\in [0,1]} with s+t=1{\displaystyle s+t=1}, f(sx+ty)≤sf(x)+tf(y){\displaystyle f(sx+ty)\leq sf(x)+tf(y)})

ExerciseProve that every continuous function f which maps [0,1] into itself has at least one fixed point, that is ∃p∈[0,1]{\displaystyle \exists p\in [0,1]} such that f(p)=p{\displaystyle f(p)=p}
Proof: Let g(x)=x−f(x){\displaystyle g(x)=x-f(x)}. Then

ExerciseProve that the space of continuous functions on an interval has the cardinality of R{\displaystyle \mathbb {R} }

ExerciseSuppose f{\displaystyle f} is defined on the set of positive real numbers and has the property: f(xy)=f(x)+f(y){\displaystyle f(xy)=f(x)+f(y)}. Then f{\displaystyle f} is unique and is a logarithm.